Use the information provided to write the standard form equation of each parabola.

**Problem 1 : **

Vertex at origin, Focus (0, 1)

**Problem 2 : **

Vertex at origin, Focus (2, 0)

**Problem 3 : **

Vertex at (1, 2), Focus (1, -1)

**Problem 4 : **

Vertex at (2, -1), Focus (-1, -1)

**Problem 5 : **

Opens up or down, Vertex at origin, Passes through (5, 75)

**Problem 6 : **

Opens left or right, Vertex (0,0), Passes through (-16, 2)

**Problem 7 : **

Opens up or down, Vertex (3, 1), Passes through (1, 9)

**Problem 8 : **

Opens left or right, Vertex (-1, -2), Passes through (11, 0)

**Problem 9 : **

Write the intercept form equation of the parabola shown below.

Use the information provided to write the standard form equation of each parabola.

**Problem 10 : **

Vertex (2, 3), Directrix : y = 6

**Problem 11 : **

Vertex (2, -2), Latus rectum : x = 1

**Problem 12 : **

Opens up, Vertex (-3, 4), Length of Latus rectum : 8 units

Use the information provided to write the standard form equation of each parabola.

**Problem 1 : **

Vertex at origin, Focus (0, 1)

**Solution : **

Plot the vertex (0, 0) and focus (0, 1) on the xy-plane.

The parabola is open up with vertex at origin.

Standard form equation of a parabola that opens up with vertex at origin :

x^{2} = 4ay

Distance between the vertex and focus is 1 unit.

That is, a = 1.

x^{2} = 4(1)y

x^{2} = 4y

**Problem 2 : **

Vertex at origin, Focus (2, 0)

**Solution : **

Plot the vertex (0, 0) and focus (2, 0) on the xy-plane.

The parabola is open to the right with vertex at origin.

Standard form equation of a parabola that opens right with vertex at origin :

y^{2} = 4ax

Distance between the vertex and focus is 2 units.

That is, a = 2.

y^{2} = 4(2)x

y^{2} = 8x

**Problem 3 : **

Vertex at (1, 2), Focus (1, -1)

**Solution : **

Plot the vertex (1, 2) and focus (1, -1) on the xy-plane.

The parabola is open down with vertex at (1, 2).

Standard form equation of a parabola that opens down with vertex at (h, k) :

(x - k)^{2} = -4a(y - h)

Vertex (h, k) = (1, 2).

(x - 1)^{2} = -4a(y - 2)

Distance between the vertex and focus is 3 units.

That is, a = 3.

(x - 1)^{2} = -4(3)(y - 2)

(x - 1)^{2} = -12(y - 2)

**Problem 4 : **

Vertex at (2, -1), Focus (-1, -1)

**Solution : **

Plot the vertex (2, -1) and focus (-1, -1) on the xy-plane.

The parabola is open to the left with vertex at (2, -1).

Standard equation of a parabola that opens left with vertex at (h, k) :

(y - k)^{2} = -4a(x - h)

Vertex (h, k) = (2, -1).

(y + 1)^{2} = -4a(x - 2)

Distance between the vertex and focus is 3 units.

That is, a = 3.

(y + 1)^{2} = -4(3)(x - 2)

(y + 1)^{2} = -12(x - 2)

Use the information provided to write the vertex form equation of each parabola.

**Problem 5 : **

Opens up or down, Vertex at origin, Passes through (5, 75)

**Solution : **

Vertex form equation of a parabola that opens up or down with vertex at origin :

y = ax^{2}

It passes through (5, 75). Substitute (x, y) = (5, 75).

75 = a(5)^{2}

75 = 25a

Divide each side by 25.

3 = a

Vertex form equation of the parabola :

y = 3x^{2}

**Problem 6 : **

Opens left or right, Vertex (0,0), Passes through (-16, 2)

**Solution : **

Vertex form equation of a parabola that opens left or right with vertex at origin :

x = ay^{2}

It passes through (-16, 2). Substitute (x, y) = (-16, 2).

-16 = a(2)^{2}

-16 = a(4)

Divide each side by 4.

-4 = a

Vertex form equation of the parabola :

x = -4y^{2}

**Problem 7 : **

Opens up or down, Vertex (3, 1), Passes through (1, 9)

**Solution : **

Vertex form equation of a parabola that opens up or down with vertex at (h, k) :

y = a(x - h)^{2} + k

Vertex (h, k) = (3, 1).

y = a(x - 3)^{2} + 1

It passes through (1, 9). Substitute (x, y) = (1, 9).

9 = a(1 - 3)^{2} + 1

9 = a(-2)^{2} + 1

9 = 4a + 1

Subtract 1 from each side.

8 = 4a

Divide each side by 4.

2 = a

Vertex form equation of the parabola :

y = 2(x - 3)^{2} + 1

**Problem 8 : **

Opens left or right, Vertex (-1, -2), Passes through (11, 0)

**Solution : **

Vertex form equation of a parabola that opens left or right with vertex at (h, k) :

x = a(y - k)^{2} + h

Vertex (h, k) = (-1, -2).

x = a(y + 2)^{2} - 1

It passes through (11, 0). Substitute (x, y) = (11, 0).

11 = a(0 + 2)^{2} - 1

11 = a(2)^{2} - 1

11 = 4a - 1

Add 1 to each side.

12 = 4a

Divide each side by 4.

3 = a

Vertex form equation of the parabola :

x = 3(y + 2)^{2} - 1

**Problem 9 : **

Write the intercept form equation of the parabola shown below.

**Solution : **

Intercept form equation of the above parabola :

y = a(x - p)(x - q)

Because x-intercepts are (-1, 0) and (2, 0),

x = -1 -----> x + 1 = 0

x = 2 -----> x - 2 = 0

Then,

y = a(x + 1)(x - 2)

It passes through (0, -4). Substitute (x, y) = (0, -4).

-4 = a(0 + 1)(0 - 2)

-4 = a(1)(-2)

-4 = -2a

Divide each side by -2.

2 = a

Intercept form equation of the parabola :

y = 2(x + 1)(x - 2)

Use the information provided to write the standard form equation of each parabola.

**Problem 10 : **

Vertex (2, 3), Directrix : y = 6

**Solution : **

Plot the vertex (2, 3) and directrix y = 6 on the xy-plane.

The parabola is open down with vertex at (2, 3).

Standard equation of a parabola that opens down with vertex at (h, k) :

(x - h)^{2} = -4a(y - k)

Vertex (h, k) = (2, 3).

(x - 2)^{2} = -4a(y - 3)

Distance between the directrix and vertex is 3 units.

That is, a = 3.

(x - 2)^{2} = -4(3)(y - 3)

(x - 2)^{2} = -12(y - 3)

**Problem 11 : **

Vertex (2, -2), Latus rectum : x = 1

**Solution : **

Plot the vertex (2, -2) and latus rectum x = 1 on the xy-plane.

The parabola is open left with vertex at (2, -2).

Standard equation of a parabola that opens left with vertex at (h, k) :

(y - k)^{2} = -4a(x - h)

Vertex (h, k) = (2, -2).

(y + 2)^{2} = -4a(x - 2)

Distance between the latus rectum and vertex is 1 unit.

That is, a = 1.

(y + 2)^{2} = -4(1)(x - 2)

(y + 2)^{2} = -4(x - 2)

**Problem 12 : **

Opens up, Vertex (-3, 4), Length of Latus rectum : 8 units

**Solution : **

Standard equation of a parabola that opens up with vertex at (h, k) :

(x - h)^{2} = 4a(y - k)

Vertex (h, k) = (-3, 4).

(x + 3)^{2} = 4a(y - 4) -----(1)

Length of Latus rectum : 8.

4a = 8

Divide each side by 4.

a = 2

Substitute 2 for a in (1).

(x + 3)^{2} = 4(2)(y - 4)

(x + 3)^{2} = 8(y - 4)

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